Let Гr,n—r denote the infimum of all number Г &gt; 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c1; c2,…, cn, there exist integersx1,x2,…,xn satisfying 0 &lt; Q(x1+c1,x2 + c2,…,xn + cn) ≤(Г¦Z &gt; ¦)1/n. All the values of Гr,n—r are known except for г1,4. Earlier it was shown that 8 ≤Г1,4 ≤16. Here we improve the upper bound to get Г1,4 &lt; 12.

We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo one, are uniformly distributed on the a-adic integers.

An important theorem of Shiu gives a (precise) bound for the average of values of multiplicative functions, of a certain class, over ‘short’ intervals. Here we obtain, by simple means, the above result of same qualitative order.

Chaotic sequences generated by nonlinear difference systems or ‘maps’ where the defining nonlinearities are polynomials, have been examined from the point of view of the sequential points seeking zeroes of an unknown functionf following the rule of Newton iterations. Following such nonlinear transformation rule, alternative sequences have been constructed showing monotonie convergence. Evidently, these are maps of the original sequences. For second degree systems, another kind of possibly less chaotic sequences have been constructed by essentially the same method. Finally, it is shown that the original chaotic system can be decomposed into a fast monotonically convergent part and a principal oscillatory part showing sharp oscillations. The methods are exemplified by the well-known logistic map, delayed-logistic map and the Hénon map.